AUSTIN — Working to avoid a special session, state lawmakers made progress on crucial issues Saturday when negotiators said they have reached consensus on a must-pass state budget and the Texas House advanced property tax changes.

There also was buzz that the House would consider a version of the so-called bathroom bill today, addressing another issue that’s dangling as the regular session races to its May 29 end.

The packed weekend follows Lt. Gov. Dan Patrick’s threat that he would force a special session unless his priorities are addressed, including property tax reform and the bathroom bill, which would require people to use public bathrooms corresponding to the sex on their birth certificates.

Gov. Greg Abbott, the only one who can call a special session, has said he wants passage of property tax changes and a bathroom bill but that he believes that lawmakers can get the job done by the end of the regular session.

Lawmakers still have work to do. The budget for the next two years must be approved by the House and Senate after being backed by their negotiating teams, which worked into the night Saturday to finalize details.

Among high points, budget negotiators agreed on how to pay for their spending plan, which was a tight one thanks to revenues affected by the uncertain oil and gas industry and past decisions to cut taxes and set aside big money for transportation.

After tangling over whether to dip into the rainy day fund, as the House proposed, the two sides agreed to spend close to $1 billion from it on one-time items. That spending would include $75 million for the Alamo redevelopment project, the amount requested by Land Commissioner George P. Bush. All told, the budget includes $87.8 million in Alamo-related spending.

The budget also would rely on $1.8 billion that would be made available by slightly delaying the transfer of money constitutionally dedicated to transportation, pushing the expense to the next fiscal cycle. That fiscal maneuver, proposed by the Senate, initially got harsh pushback from House leaders.

Among other provisions, the budget agreement would provide more funding for Abbott priorities that had been given short-shrift, including

$236 million he wanted for his initiative on high-quality prekindergarten. That proposal initially got no money from either chamber.

Negotiators also opted against a dramatic change in higher education funding that had been proposed by the Senate targeting so-called special items that are paid for outside of formulas. The agreement would continue special-item funding for two years while higher education funding is studied. Special items characterized as startups would be cut by 50 percent if they’d been in existence for more than eight years, in order to make the point that special items aren’t supposed to last forever.

There also is more action needed on the tax measure, which requires another House vote before going to the Senate for consideration.

The House isn’t poised to deliver all the tax changes Patrick has sought.

It gave preliminary approval to a proposal that is meant to give taxpayers more information and empower them to press for changes if they dislike a proposed tax rate.

But it didn’t vote on automatic rollback elections that the Senate has approved under Patrick and that are opposed by cities and counties.

“There weren’t votes in the House to pass a rollback change,” said House Ways and Means Committee Chairman Dennis Bonnen, R-Angleton. He attached his transparency provisions to another Senate tax bill with 136-0 approval in the House.

Similarly, some have said a bathroom proposal considered in that chamber could be restricted to schools. Patrick, calling it a matter of public safety and privacy, earlier championed Senate passage of a measure to specify that people must use restrooms, dressing rooms and locker rooms in government buildings, schools and universities that align with the sex on their birth certificates.

“The lieutenant governor has used his leverage effectively, and he is watching the House and the action they are taking on the key issues he identified (last) week including property taxes, privacy, life, school choice and school finance, first responder and veterans issues and photo voter ID. The lieutenant governor has already been successful in his goal of a conservative budget that does not use rainy day funds for ongoing expenses,” Garcia said.

Bonnen said the provisions benefit taxpayers.

“It’ll give them a level of information and transparency and clarity on property taxes that has never, ever existed, and it will empower them to advocate with their local governments on their taxes with information they’ve never had before,” Bonnen said.

His proposal includes provisions aimed at ensuring that taxpayers know whether their bills will increase under a proposed tax rate, by how much and which local taxing entities are responsible, allowing them a better opportunity to press for changes if they’re not satisfied.

Cities and counties, which had argued that automatic rollback elections would impair their ability to properly fund services including public safety, were satisfied.

“We are grateful that the House did the right thing: reform the appraisal process without putting public safety, infrastructure and other much-needed local community priorities at risk,” said Jeff Coyle, director of government and public affairs for the city of San Antonio.

Bexar County Judge Nelson Wolff said, “Hopefully, the Senate will concur in it (the House action), and things will come out better for taxpayers as well as allow us to make the decisions we’ve got to make … as we struggle with all the issues we’ve got to face.”

Paul Sugg, legislative director of the Texas Association of Counties, said, “We support transparency throughout the entire process. Having more citizens engaged in what counties are doing when they’re setting a tax rate or adopting a budget is critically important. This is what representative government is about.”

Also Saturday, the House took action that could address one way Patrick could force a special session over his priority issues.

Patrick claimed the leverage by saying he wouldn’t allow Senate action on a so-called safety net bill for state agencies under sunset review until his demands were met.

Agencies under review are put on a course to be abolished unless lawmakers vote to continue them. The safety net bill would save agencies that don’t get a legislative vote in the regular session.

But the House amended another bill with the safety net provisions, which would allow the agencies to continue even if the main bill doesn’t pass. The Senate still would have to agree to the House amendment.